Newspace parameters
| Level: | \( N \) | \(=\) | \( 459 = 3^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 459.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.66513345278\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.404.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 5x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 5x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.74483 | 0 | 5.53407 | −1.53407 | 0 | −4.95558 | −9.70041 | 0 | 4.21076 | |||||||||||||||||||||||||||
| 1.2 | 1.39593 | 0 | −0.0513742 | 4.05137 | 0 | −2.25951 | −2.86358 | 0 | 5.65544 | ||||||||||||||||||||||||||||
| 1.3 | 2.34889 | 0 | 3.51730 | 0.482696 | 0 | 3.21509 | 3.56399 | 0 | 1.13380 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(17\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 459.2.a.n | yes | 3 |
| 3.b | odd | 2 | 1 | 459.2.a.m | ✓ | 3 | |
| 4.b | odd | 2 | 1 | 7344.2.a.cn | 3 | ||
| 12.b | even | 2 | 1 | 7344.2.a.ci | 3 | ||
| 17.b | even | 2 | 1 | 7803.2.a.bg | 3 | ||
| 51.c | odd | 2 | 1 | 7803.2.a.bd | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 459.2.a.m | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 459.2.a.n | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 7344.2.a.ci | 3 | 12.b | even | 2 | 1 | ||
| 7344.2.a.cn | 3 | 4.b | odd | 2 | 1 | ||
| 7803.2.a.bd | 3 | 51.c | odd | 2 | 1 | ||
| 7803.2.a.bg | 3 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(459))\):
|
\( T_{2}^{3} - T_{2}^{2} - 7T_{2} + 9 \)
|
|
\( T_{5}^{3} - 3T_{5}^{2} - 5T_{5} + 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - T^{2} - 7T + 9 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 3 T^{2} + \cdots + 3 \)
|
| $7$ |
\( T^{3} + 4 T^{2} + \cdots - 36 \)
|
| $11$ |
\( T^{3} - 4 T^{2} + \cdots + 48 \)
|
| $13$ |
\( T^{3} - T^{2} + \cdots + 29 \)
|
| $17$ |
\( (T + 1)^{3} \)
|
| $19$ |
\( T^{3} - 5 T^{2} + \cdots + 9 \)
|
| $23$ |
\( T^{3} - 3 T^{2} + \cdots + 63 \)
|
| $29$ |
\( T^{3} + 3 T^{2} + \cdots - 3 \)
|
| $31$ |
\( T^{3} - 32T + 32 \)
|
| $37$ |
\( T^{3} - 12 T^{2} + \cdots - 28 \)
|
| $41$ |
\( T^{3} + 3 T^{2} + \cdots - 3 \)
|
| $43$ |
\( T^{3} - 5 T^{2} + \cdots + 9 \)
|
| $47$ |
\( T^{3} + 4 T^{2} + \cdots + 132 \)
|
| $53$ |
\( T^{3} - 4 T^{2} + \cdots - 84 \)
|
| $59$ |
\( T^{3} + 30 T^{2} + \cdots + 924 \)
|
| $61$ |
\( T^{3} - 36 T^{2} + \cdots - 1636 \)
|
| $67$ |
\( T^{3} + 29 T^{2} + \cdots + 751 \)
|
| $71$ |
\( T^{3} + 19 T^{2} + \cdots - 111 \)
|
| $73$ |
\( T^{3} - 200T + 500 \)
|
| $79$ |
\( T^{3} - 14 T^{2} + \cdots - 36 \)
|
| $83$ |
\( T^{3} - 4 T^{2} + \cdots + 672 \)
|
| $89$ |
\( T^{3} - 30 T^{2} + \cdots - 648 \)
|
| $97$ |
\( T^{3} - 8T - 4 \)
|
| show more | |
| show less |